I don't think you'd ever be able to use the Loopback audio port (which returns the entire output signal - back to an input port to allow recording) and be able to process the resultant input signal while monitoring the results.
That scenario is the very definition of a feedback loop. ;)
I'd just save the processing on the returning (input) signal for post production.
That's where you're going to assemble/edit the video.
If I'm understanding properly, that would be the perfect time to "sweeten" the audio.
Otherwise, you could use external devices (probably not what you're looking for)
ie: Process the audio output with external gear... and capture the processed audio back into the DAW (or using a portable recorder - to capture the audio). Separate portable audio recorders are often used when dealing with lower budget video shoots... where the onboard audio quality is sketchy.
What about using the UR-44's onboard DSP to process the final output?
If you could get by with those processing tools, you could hear the result without creating feedback (and you could record the result using the Loopback port). Essentially using "external" (to the DAW) processing...